47邊形（二）
續(xù)《47邊形（一）》
二．用22個含有23次單位根的23次根式線性表示sin(2*n*pi/47)。以2為底數。
F0=2*(sin(2*pi/47)+sin(4*pi/47)+sin(8*pi/47)+sin(16*pi/47)+sin(32*pi/47)+sin(64*pi/47)+sin(34*pi/47)+sin(68*pi/47)+sin(42*pi/47)+sin(84*pi/47)+sin(74*pi/47)+sin(54*pi/47)+sin(14*pi/47)+sin(28*pi/47)+sin(56*pi/47)+sin(18*pi/47)+sin(36*pi/47)+sin(72*pi/47)+sin(50*pi/47)+sin(6*pi/47)+sin(12*pi/47)+sin(24*pi/47)+sin(48*pi/47))=sqrt(47);
F1=2*(sin(2*pi/47)+k1*sin(4*pi/47)+k2*sin(8*pi/47)+k3*sin(16*pi/47)+k4*sin(32*pi/47)+k5*sin(64*pi/47)+k6*sin(34*pi/47)+k7*sin(68*pi/47)+k8*sin(42*pi/47)+k9*sin(84*pi/47)+k10*sin(74*pi/47)+k11*sin(54*pi/47)+k12*sin(14*pi/47)+k13*sin(28*pi/47)+k14*sin(56*pi/47)+k15*sin(18*pi/47)+k16*sin(36*pi/47)+k17*sin(72*pi/47)+k18*sin(50*pi/47)+k19*sin(6*pi/47)+k20*sin(12*pi/47)+k21*sin(24*pi/47)+k22*sin(48*pi/47));
F2=2*(sin(2*pi/47)+k2*sin(4*pi/47)+k4*sin(8*pi/47)+k6*sin(16*pi/47)+k8*sin(32*pi/47)+k10*sin(64*pi/47)+k12*sin(34*pi/47)+k14*sin(68*pi/47)+k16*sin(42*pi/47)+k18*sin(84*pi/47)+k20*sin(74*pi/47)+k22*sin(54*pi/47)+k1*sin(14*pi/47)+k3*sin(28*pi/47)+k5*sin(56*pi/47)+k7*sin(18*pi/47)+k9*sin(36*pi/47)+k11*sin(72*pi/47)+k13*sin(50*pi/47)+k15*sin(6*pi/47)+k17*sin(12*pi/47)+k19*sin(24*pi/47)+k21*sin(48*pi/47));
F3=2*(sin(2*pi/47)+k3*sin(4*pi/47)+k6*sin(8*pi/47)+k9*sin(16*pi/47)+k12*sin(32*pi/47)+k15*sin(64*pi/47)+k18*sin(34*pi/47)+k21*sin(68*pi/47)+k1*sin(42*pi/47)+k4*sin(84*pi/47)+k7*sin(74*pi/47)+k10*sin(54*pi/47)+k13*sin(14*pi/47)+k16*sin(28*pi/47)+k19*sin(56*pi/47)+k22*sin(18*pi/47)+k2*sin(36*pi/47)+k5*sin(72*pi/47)+k8*sin(50*pi/47)+k11*sin(6*pi/47)+k14*sin(12*pi/47)+k17*sin(24*pi/47)+k20*sin(48*pi/47));
F4=2*(sin(2*pi/47)+k4*sin(4*pi/47)+k8*sin(8*pi/47)+k12*sin(16*pi/47)+k16*sin(32*pi/47)+k20*sin(64*pi/47)+k1*sin(34*pi/47)+k5*sin(68*pi/47)+k9*sin(42*pi/47)+k13*sin(84*pi/47)+k17*sin(74*pi/47)+k21*sin(54*pi/47)+k2*sin(14*pi/47)+k6*sin(28*pi/47)+k10*sin(56*pi/47)+k14*sin(18*pi/47)+k18*sin(36*pi/47)+k22*sin(72*pi/47)+k3*sin(50*pi/47)+k7*sin(6*pi/47)+k11*sin(12*pi/47)+k15*sin(24*pi/47)+k19*sin(48*pi/47));
F5=2*(sin(2*pi/47)+k5*sin(4*pi/47)+k10*sin(8*pi/47)+k15*sin(16*pi/47)+k20*sin(32*pi/47)+k2*sin(64*pi/47)+k7*sin(34*pi/47)+k12*sin(68*pi/47)+k17*sin(42*pi/47)+k22*sin(84*pi/47)+k4*sin(74*pi/47)+k9*sin(54*pi/47)+k14*sin(14*pi/47)+k19*sin(28*pi/47)+k1*sin(56*pi/47)+k6*sin(18*pi/47)+k11*sin(36*pi/47)+k16*sin(72*pi/47)+k21*sin(50*pi/47)+k3*sin(6*pi/47)+k8*sin(12*pi/47)+k13*sin(24*pi/47)+k18*sin(48*pi/47));
F6=2*(sin(2*pi/47)+k6*sin(4*pi/47)+k12*sin(8*pi/47)+k18*sin(16*pi/47)+k1*sin(32*pi/47)+k7*sin(64*pi/47)+k13*sin(34*pi/47)+k19*sin(68*pi/47)+k2*sin(42*pi/47)+k8*sin(84*pi/47)+k14*sin(74*pi/47)+k20*sin(54*pi/47)+k3*sin(14*pi/47)+k9*sin(28*pi/47)+k15*sin(56*pi/47)+k21*sin(18*pi/47)+k4*sin(36*pi/47)+k10*sin(72*pi/47)+k16*sin(50*pi/47)+k22*sin(6*pi/47)+k5*sin(12*pi/47)+k11*sin(24*pi/47)+k17*sin(48*pi/47));
F7=2*(sin(2*pi/47)+k7*sin(4*pi/47)+k14*sin(8*pi/47)+k21*sin(16*pi/47)+k5*sin(32*pi/47)+k12*sin(64*pi/47)+k19*sin(34*pi/47)+k3*sin(68*pi/47)+k10*sin(42*pi/47)+k17*sin(84*pi/47)+k1*sin(74*pi/47)+k8*sin(54*pi/47)+k15*sin(14*pi/47)+k22*sin(28*pi/47)+k6*sin(56*pi/47)+k13*sin(18*pi/47)+k20*sin(36*pi/47)+k4*sin(72*pi/47)+k11*sin(50*pi/47)+k18*sin(6*pi/47)+k2*sin(12*pi/47)+k9*sin(24*pi/47)+k16*sin(48*pi/47));
F8=2*(sin(2*pi/47)+k8*sin(4*pi/47)+k16*sin(8*pi/47)+k1*sin(16*pi/47)+k9*sin(32*pi/47)+k17*sin(64*pi/47)+k2*sin(34*pi/47)+k10*sin(68*pi/47)+k18*sin(42*pi/47)+k3*sin(84*pi/47)+k11*sin(74*pi/47)+k19*sin(54*pi/47)+k4*sin(14*pi/47)+k12*sin(28*pi/47)+k20*sin(56*pi/47)+k5*sin(18*pi/47)+k13*sin(36*pi/47)+k21*sin(72*pi/47)+k6*sin(50*pi/47)+k14*sin(6*pi/47)+k22*sin(12*pi/47)+k7*sin(24*pi/47)+k15*sin(48*pi/47));
F9=2*(sin(2*pi/47)+k9*sin(4*pi/47)+k18*sin(8*pi/47)+k4*sin(16*pi/47)+k13*sin(32*pi/47)+k22*sin(64*pi/47)+k8*sin(34*pi/47)+k17*sin(68*pi/47)+k3*sin(42*pi/47)+k12*sin(84*pi/47)+k21*sin(74*pi/47)+k7*sin(54*pi/47)+k16*sin(14*pi/47)+k2*sin(28*pi/47)+k11*sin(56*pi/47)+k20*sin(18*pi/47)+k6*sin(36*pi/47)+k15*sin(72*pi/47)+k1*sin(50*pi/47)+k10*sin(6*pi/47)+k19*sin(12*pi/47)+k5*sin(24*pi/47)+k14*sin(48*pi/47));
F10=2*(sin(2*pi/47)+k10*sin(4*pi/47)+k20*sin(8*pi/47)+k7*sin(16*pi/47)+k17*sin(32*pi/47)+k4*sin(64*pi/47)+k14*sin(34*pi/47)+k1*sin(68*pi/47)+k11*sin(42*pi/47)+k21*sin(84*pi/47)+k8*sin(74*pi/47)+k18*sin(54*pi/47)+k5*sin(14*pi/47)+k15*sin(28*pi/47)+k2*sin(56*pi/47)+k12*sin(18*pi/47)+k22*sin(36*pi/47)+k9*sin(72*pi/47)+k19*sin(50*pi/47)+k6*sin(6*pi/47)+k16*sin(12*pi/47)+k3*sin(24*pi/47)+k13*sin(48*pi/47));
F11=2*(sin(2*pi/47)+k11*sin(4*pi/47)+k22*sin(8*pi/47)+k10*sin(16*pi/47)+k21*sin(32*pi/47)+k9*sin(64*pi/47)+k20*sin(34*pi/47)+k8*sin(68*pi/47)+k19*sin(42*pi/47)+k7*sin(84*pi/47)+k18*sin(74*pi/47)+k6*sin(54*pi/47)+k17*sin(14*pi/47)+k5*sin(28*pi/47)+k16*sin(56*pi/47)+k4*sin(18*pi/47)+k15*sin(36*pi/47)+k3*sin(72*pi/47)+k14*sin(50*pi/47)+k2*sin(6*pi/47)+k13*sin(12*pi/47)+k1*sin(24*pi/47)+k12*sin(48*pi/47));
F12=47/F11; F13=47/F10; F14=47/F9; F15=47/F8; F16=47/F7; F17=47/F6; F18=47/F5; F19=47/F4;
F20=47/F3; F21=47/F2; F22=47/F1;
Y1=F1^23; Y2=F2^23;Y3=F3^23; Y4=F4^23; Y5=F5^23; Y6=F6^23; Y7=F7^23; Y8=F8^23;
Y9=F9^23; Y10=F10^23; Y11=F11^23; Y12=F12^23; Y13=F13^23; Y14=F14^23; Y15=F15^23; Y16=F16^23;
Y17=F17^23; Y18=F18^23; Y19=F19^23; Y20=F20^23; Y21=F21^23; Y22=F22^23; Y0=47^11.5;
假設Y1~Y22能用23次單位根線性表示，表達式為Yn=(D0+D1*k1^n+……+D22*k22^n)*sqrt(47)，D0~D22都是龐大的整數！
D0=(Y1+Y2+Y3+Y4+Y5+Y6+Y7+Y8+Y9+Y10+Y11+Y12+Y13+Y14+Y15+Y16+Y17+Y18+Y19+Y20+Y21+Y22+Y0)/sqrt(24863);
D1=(Y1/k1^1+Y2/k2^1+Y3/k3^1+Y4/k4^1+Y5/k5^1+Y6/k6^1+Y7/k7^1+Y8/k8^1+Y9/k9^1+Y10/k10^1+Y11/k11^1+Y12/k12^1+Y13/k13^1+Y14/k14^1+Y15/k15^1+Y16/k16^1+Y17/k17^1+Y18/k18^1+Y19/k19^1+Y20/k20^1+Y21/k21^1+Y22/k22^1+Y0)/sqrt(24863);
D2=(Y1/k1^2+Y2/k2^2+Y3/k3^2+Y4/k4^2+Y5/k5^2+Y6/k6^2+Y7/k7^2+Y8/k8^2+Y9/k9^2+Y10/k10^2+Y11/k11^2+Y12/k12^2+Y13/k13^2+Y14/k14^2+Y15/k15^2+Y16/k16^2+Y17/k17^2+Y18/k18^2+Y19/k19^2+Y20/k20^2+Y21/k21^2+Y22/k22^2+Y0)/sqrt(24863);
D3=(Y1/k1^3+Y2/k2^3+Y3/k3^3+Y4/k4^3+Y5/k5^3+Y6/k6^3+Y7/k7^3+Y8/k8^3+Y9/k9^3+Y10/k10^3+Y11/k11^3+Y12/k12^3+Y13/k13^3+Y14/k14^3+Y15/k15^3+Y16/k16^3+Y17/k17^3+Y18/k18^3+Y19/k19^3+Y20/k20^3+Y21/k21^3+Y22/k22^3+Y0)/sqrt(24863);
D4=(Y1/k1^4+Y2/k2^4+Y3/k3^4+Y4/k4^4+Y5/k5^4+Y6/k6^4+Y7/k7^4+Y8/k8^4+Y9/k9^4+Y10/k10^4+Y11/k11^4+Y12/k12^4+Y13/k13^4+Y14/k14^4+Y15/k15^4+Y16/k16^4+Y17/k17^4+Y18/k18^4+Y19/k19^4+Y20/k20^4+Y21/k21^4+Y22/k22^4+Y0)/sqrt(24863);
D5=(Y1/k1^5+Y2/k2^5+Y3/k3^5+Y4/k4^5+Y5/k5^5+Y6/k6^5+Y7/k7^5+Y8/k8^5+Y9/k9^5+Y10/k10^5+Y11/k11^5+Y12/k12^5+Y13/k13^5+Y14/k14^5+Y15/k15^5+Y16/k16^5+Y17/k17^5+Y18/k18^5+Y19/k19^5+Y20/k20^5+Y21/k21^5+Y22/k22^5+Y0)/sqrt(24863);
D6=(Y1/k1^6+Y2/k2^6+Y3/k3^6+Y4/k4^6+Y5/k5^6+Y6/k6^6+Y7/k7^6+Y8/k8^6+Y9/k9^6+Y10/k10^6+Y11/k11^6+Y12/k12^6+Y13/k13^6+Y14/k14^6+Y15/k15^6+Y16/k16^6+Y17/k17^6+Y18/k18^6+Y19/k19^6+Y20/k20^6+Y21/k21^6+Y22/k22^6+Y0)/sqrt(24863);
D7=(Y1/k1^7+Y2/k2^7+Y3/k3^7+Y4/k4^7+Y5/k5^7+Y6/k6^7+Y7/k7^7+Y8/k8^7+Y9/k9^7+Y10/k10^7+Y11/k11^7+Y12/k12^7+Y13/k13^7+Y14/k14^7+Y15/k15^7+Y16/k16^7+Y17/k17^7+Y18/k18^7+Y19/k19^7+Y20/k20^7+Y21/k21^7+Y22/k22^7+Y0)/sqrt(24863);
D8=(Y1/k1^8+Y2/k2^8+Y3/k3^8+Y4/k4^8+Y5/k5^8+Y6/k6^8+Y7/k7^8+Y8/k8^8+Y9/k9^8+Y10/k10^8+Y11/k11^8+Y12/k12^8+Y13/k13^8+Y14/k14^8+Y15/k15^8+Y16/k16^8+Y17/k17^8+Y18/k18^8+Y19/k19^8+Y20/k20^8+Y21/k21^8+Y22/k22^8+Y0)/sqrt(24863);
D9=(Y1/k1^9+Y2/k2^9+Y3/k3^9+Y4/k4^9+Y5/k5^9+Y6/k6^9+Y7/k7^9+Y8/k8^9+Y9/k9^9+Y10/k10^9+Y11/k11^9+Y12/k12^9+Y13/k13^9+Y14/k14^9+Y15/k15^9+Y16/k16^9+Y17/k17^9+Y18/k18^9+Y19/k19^9+Y20/k20^9+Y21/k21^9+Y22/k22^9+Y0)/sqrt(24863);
D10=(Y1/k1^10+Y2/k2^10+Y3/k3^10+Y4/k4^10+Y5/k5^10+Y6/k6^10+Y7/k7^10+Y8/k8^10+Y9/k9^10+Y10/k10^10+Y11/k11^10+Y12/k12^10+Y13/k13^10+Y14/k14^10+Y15/k15^10+Y16/k16^10+Y17/k17^10+Y18/k18^10+Y19/k19^10+Y20/k20^10+Y21/k21^10+Y22/k22^10+Y0)/sqrt(24863);
D11=(Y1/k1^11+Y2/k2^11+Y3/k3^11+Y4/k4^11+Y5/k5^11+Y6/k6^11+Y7/k7^11+Y8/k8^11+Y9/k9^11+Y10/k10^11+Y11/k11^11+Y12/k12^11+Y13/k13^11+Y14/k14^11+Y15/k15^11+Y16/k16^11+Y17/k17^11+Y18/k18^11+Y19/k19^11+Y20/k20^11+Y21/k21^11+Y22/k22^11+Y0)/sqrt(24863);
D12=(Y1/k1^12+Y2/k2^12+Y3/k3^12+Y4/k4^12+Y5/k5^12+Y6/k6^12+Y7/k7^12+Y8/k8^12+Y9/k9^12+Y10/k10^12+Y11/k11^12+Y12/k12^12+Y13/k13^12+Y14/k14^12+Y15/k15^12+Y16/k16^12+Y17/k17^12+Y18/k18^12+Y19/k19^12+Y20/k20^12+Y21/k21^12+Y22/k22^12+Y0)/sqrt(24863);
D13=(Y1/k1^13+Y2/k2^13+Y3/k3^13+Y4/k4^13+Y5/k5^13+Y6/k6^13+Y7/k7^13+Y8/k8^13+Y9/k9^13+Y10/k10^13+Y11/k11^13+Y12/k12^13+Y13/k13^13+Y14/k14^13+Y15/k15^13+Y16/k16^13+Y17/k17^13+Y18/k18^13+Y19/k19^13+Y20/k20^13+Y21/k21^13+Y22/k22^13+Y0)/sqrt(24863);
D14=(Y1/k1^14+Y2/k2^14+Y3/k3^14+Y4/k4^14+Y5/k5^14+Y6/k6^14+Y7/k7^14+Y8/k8^14+Y9/k9^14+Y10/k10^14+Y11/k11^14+Y12/k12^14+Y13/k13^14+Y14/k14^14+Y15/k15^14+Y16/k16^14+Y17/k17^14+Y18/k18^14+Y19/k19^14+Y20/k20^14+Y21/k21^14+Y22/k22^14+Y0)/sqrt(24863);
D15=(Y1/k1^15+Y2/k2^15+Y3/k3^15+Y4/k4^15+Y5/k5^15+Y6/k6^15+Y7/k7^15+Y8/k8^15+Y9/k9^15+Y10/k10^15+Y11/k11^15+Y12/k12^15+Y13/k13^15+Y14/k14^15+Y15/k15^15+Y16/k16^15+Y17/k17^15+Y18/k18^15+Y19/k19^15+Y20/k20^15+Y21/k21^15+Y22/k22^15+Y0)/sqrt(24863);
D16=(Y1/k1^16+Y2/k2^16+Y3/k3^16+Y4/k4^16+Y5/k5^16+Y6/k6^16+Y7/k7^16+Y8/k8^16+Y9/k9^16+Y10/k10^16+Y11/k11^16+Y12/k12^16+Y13/k13^16+Y14/k14^16+Y15/k15^16+Y16/k16^16+Y17/k17^16+Y18/k18^16+Y19/k19^16+Y20/k20^16+Y21/k21^16+Y22/k22^16+Y0)/sqrt(24863);
D17=(Y1/k1^17+Y2/k2^17+Y3/k3^17+Y4/k4^17+Y5/k5^17+Y6/k6^17+Y7/k7^17+Y8/k8^17+Y9/k9^17+Y10/k10^17+Y11/k11^17+Y12/k12^17+Y13/k13^17+Y14/k14^17+Y15/k15^17+Y16/k16^17+Y17/k17^17+Y18/k18^17+Y19/k19^17+Y20/k20^17+Y21/k21^17+Y22/k22^17+Y0)/sqrt(24863);
D18=(Y1/k1^18+Y2/k2^18+Y3/k3^18+Y4/k4^18+Y5/k5^18+Y6/k6^18+Y7/k7^18+Y8/k8^18+Y9/k9^18+Y10/k10^18+Y11/k11^18+Y12/k12^18+Y13/k13^18+Y14/k14^18+Y15/k15^18+Y16/k16^18+Y17/k17^18+Y18/k18^18+Y19/k19^18+Y20/k20^18+Y21/k21^18+Y22/k22^18+Y0)/sqrt(24863);
D19=(Y1/k1^19+Y2/k2^19+Y3/k3^19+Y4/k4^19+Y5/k5^19+Y6/k6^19+Y7/k7^19+Y8/k8^19+Y9/k9^19+Y10/k10^19+Y11/k11^19+Y12/k12^19+Y13/k13^19+Y14/k14^19+Y15/k15^19+Y16/k16^19+Y17/k17^19+Y18/k18^19+Y19/k19^19+Y20/k20^19+Y21/k21^19+Y22/k22^19+Y0)/sqrt(24863);
D20=(Y1/k1^20+Y2/k2^20+Y3/k3^20+Y4/k4^20+Y5/k5^20+Y6/k6^20+Y7/k7^20+Y8/k8^20+Y9/k9^20+Y10/k10^20+Y11/k11^20+Y12/k12^20+Y13/k13^20+Y14/k14^20+Y15/k15^20+Y16/k16^20+Y17/k17^20+Y18/k18^20+Y19/k19^20+Y20/k20^20+Y21/k21^20+Y22/k22^20+Y0)/sqrt(24863);
D21=(Y1/k1^21+Y2/k2^21+Y3/k3^21+Y4/k4^21+Y5/k5^21+Y6/k6^21+Y7/k7^21+Y8/k8^21+Y9/k9^21+Y10/k10^21+Y11/k11^21+Y12/k12^21+Y13/k13^21+Y14/k14^21+Y15/k15^21+Y16/k16^21+Y17/k17^21+Y18/k18^21+Y19/k19^21+Y20/k20^21+Y21/k21^21+Y22/k22^21+Y0)/sqrt(24863);
D22=(Y1/k1^22+Y2/k2^22+Y3/k3^22+Y4/k4^22+Y5/k5^22+Y6/k6^22+Y7/k7^22+Y8/k8^22+Y9/k9^22+Y10/k10^22+Y11/k11^22+Y12/k12^22+Y13/k13^22+Y14/k14^22+Y15/k15^22+Y16/k16^22+Y17/k17^22+Y18/k18^22+Y19/k19^22+Y20/k20^22+Y21/k21^22+Y22/k22^22+Y0)/sqrt(24863);
得到以下關系式
(1)
D0-D22=284369571646197540; D1-D22=811049183285399890; D2-D22=1593585237352147985; D3-D22=1284787122351957913;
D4-D22=808867007855627582; D5-D22=1055635222926786097; D6-D22=876689391645197176;
D7-D22=599051054911901654; D8-D22=717514008288053694; D9-D22=-160090977681806429;
D10-D22=1402330178847304956; D11-D22=396563939508167961; D12-D22=612829934225357570;
D13-D22=131453150804922790; D14-D22=1510916177987207658; D15-D22=514826183356996139;
D16-D22=-274028959817533291; D17-D22=873302494907324285; D18-D22=432446621716420445;
D19-D22=550021082584940307; D20-D22=1088241091554872128; D21-D22=143212561542595904;
(2)
Y1=sqrt(47)*(284369571646197540+811049183285399890*k1+1593585237352147985*k2+1284787122351957913*k3+808867007855627582*k4+1055635222926786097*k5+876689391645197176*k6+599051054911901654*k7+717514008288053694*k8-160090977681806429*k9+1402330178847304956*k10+396563939508167961*k11+612829934225357570*k12+131453150804922790*k13+1510916177987207658*k14+514826183356996139*k15-274028959817533291*k16+873302494907324285*k17+432446621716420445*k18+550021082584940307*k19+1088241091554872128*k20+143212561542595904*k21);
?
Y2=sqrt(47)*(284369571646197540+811049183285399890*k2+1593585237352147985*k4+1284787122351957913*k6+808867007855627582*k8+1055635222926786097*k10+876689391645197176*k12+599051054911901654*k14+717514008288053694*k16-160090977681806429*k18+1402330178847304956*k20+396563939508167961*k22+612829934225357570*k1+131453150804922790*k3+1510916177987207658*k5+514826183356996139*k7-274028959817533291*k9+873302494907324285*k11+432446621716420445*k13+550021082584940307*k15+1088241091554872128*k17+143212561542595904*k19);
?
Y3=sqrt(47)*(284369571646197540+811049183285399890*k3+1593585237352147985*k6+1284787122351957913*k9+808867007855627582*k12+1055635222926786097*k15+876689391645197176*k18+599051054911901654*k21+717514008288053694*k1-160090977681806429*k4+1402330178847304956*k7+396563939508167961*k10+612829934225357570*k13+131453150804922790*k16+1510916177987207658*k19+514826183356996139*k22-274028959817533291*k2+873302494907324285*k5+432446621716420445*k8+550021082584940307*k11+1088241091554872128*k14+143212561542595904*k17);
?
Y4=sqrt(47)*(284369571646197540+811049183285399890*k4+1593585237352147985*k8+1284787122351957913*k12+808867007855627582*k16+1055635222926786097*k20+876689391645197176*k1+599051054911901654*k5+717514008288053694*k9-160090977681806429*k13+1402330178847304956*k17+396563939508167961*k21+612829934225357570*k2+131453150804922790*k6+1510916177987207658*k10+514826183356996139*k14-274028959817533291*k18+873302494907324285*k22+432446621716420445*k3+550021082584940307*k7+1088241091554872128*k11+143212561542595904*k15);
?
Y5=sqrt(47)*(284369571646197540+811049183285399890*k5+1593585237352147985*k10+1284787122351957913*k15+808867007855627582*k20+1055635222926786097*k2+876689391645197176*k7+599051054911901654*k12+717514008288053694*k17-160090977681806429*k22+1402330178847304956*k4+396563939508167961*k9+612829934225357570*k14+131453150804922790*k19+1510916177987207658*k1+514826183356996139*k6-274028959817533291*k11+873302494907324285*k16+432446621716420445*k21+550021082584940307*k3+1088241091554872128*k8+143212561542595904*k13);
?
Y6=sqrt(47)*(284369571646197540+811049183285399890*k6+1593585237352147985*k12+1284787122351957913*k18+808867007855627582*k1+1055635222926786097*k7+876689391645197176*k13+599051054911901654*k19+717514008288053694*k2-160090977681806429*k8+1402330178847304956*k14+396563939508167961*k20+612829934225357570*k3+131453150804922790*k9+1510916177987207658*k15+514826183356996139*k21-274028959817533291*k4+873302494907324285*k10+432446621716420445*k16+550021082584940307*k22+1088241091554872128*k5+143212561542595904*k11);
?
Y7=sqrt(47)*(284369571646197540+811049183285399890*k7+1593585237352147985*k14+1284787122351957913*k21+808867007855627582*k5+1055635222926786097*k12+876689391645197176*k19+599051054911901654*k3+717514008288053694*k10-160090977681806429*k17+1402330178847304956*k1+396563939508167961*k8+612829934225357570*k15+131453150804922790*k22+1510916177987207658*k6+514826183356996139*k13-274028959817533291*k20+873302494907324285*k4+432446621716420445*k11+550021082584940307*k18+1088241091554872128*k2+143212561542595904*k9);
?
Y8=sqrt(47)*(284369571646197540+811049183285399890*k8+1593585237352147985*k16+1284787122351957913*k1+808867007855627582*k9+1055635222926786097*k17+876689391645197176*k2+599051054911901654*k10+717514008288053694*k18-160090977681806429*k3+1402330178847304956*k11+396563939508167961*k19+612829934225357570*k4+131453150804922790*k12+1510916177987207658*k20+514826183356996139*k5-274028959817533291*k13+873302494907324285*k21+432446621716420445*k6+550021082584940307*k14+1088241091554872128*k22+143212561542595904*k7);
?
Y9=sqrt(47)*(284369571646197540+811049183285399890*k9+1593585237352147985*k18+1284787122351957913*k4+808867007855627582*k13+1055635222926786097*k22+876689391645197176*k8+599051054911901654*k17+717514008288053694*k3-160090977681806429*k12+1402330178847304956*k21+396563939508167961*k7+612829934225357570*k16+131453150804922790*k2+1510916177987207658*k11+514826183356996139*k20-274028959817533291*k6+873302494907324285*k15+432446621716420445*k1+550021082584940307*k10+1088241091554872128*k19+143212561542595904*k5);
?
Y10=sqrt(47)*(284369571646197540+811049183285399890*k10+1593585237352147985*k20+1284787122351957913*k7+808867007855627582*k17+1055635222926786097*k4+876689391645197176*k14+599051054911901654*k1+717514008288053694*k11-160090977681806429*k21+1402330178847304956*k8+396563939508167961*k18+612829934225357570*k5+131453150804922790*k15+1510916177987207658*k2+514826183356996139*k12-274028959817533291*k22+873302494907324285*k9+432446621716420445*k19+550021082584940307*k6+1088241091554872128*k16+143212561542595904*k3);
?
Y11=sqrt(47)*(284369571646197540+811049183285399890*k11+1593585237352147985*k22+1284787122351957913*k10+808867007855627582*k21+1055635222926786097*k9+876689391645197176*k20+599051054911901654*k8+717514008288053694*k19-160090977681806429*k7+1402330178847304956*k18+396563939508167961*k6+612829934225357570*k17+131453150804922790*k5+1510916177987207658*k16+514826183356996139*k4-274028959817533291*k15+873302494907324285*k3+432446621716420445*k14+550021082584940307*k2+1088241091554872128*k13+143212561542595904*k1);
?
Y12=sqrt(47)*(284369571646197540+811049183285399890/k11+1593585237352147985/k22+1284787122351957913/k10+808867007855627582/k21+1055635222926786097/k9+876689391645197176/k20+599051054911901654/k8+717514008288053694/k19-160090977681806429/k7+1402330178847304956/k18+396563939508167961/k6+612829934225357570/k17+131453150804922790/k5+1510916177987207658/k16+514826183356996139/k4-274028959817533291/k15+873302494907324285/k3+432446621716420445/k14+550021082584940307/k2+1088241091554872128/k13+143212561542595904/k1);
?
Y13=sqrt(47)*(284369571646197540+811049183285399890/k10+1593585237352147985/k20+1284787122351957913/k7+808867007855627582/k17+1055635222926786097/k4+876689391645197176/k14+599051054911901654/k1+717514008288053694/k11-160090977681806429/k21+1402330178847304956/k8+396563939508167961/k18+612829934225357570/k5+131453150804922790/k15+1510916177987207658/k2+514826183356996139/k12-274028959817533291/k22+873302494907324285/k9+432446621716420445/k19+550021082584940307/k6+1088241091554872128/k16+143212561542595904/k3);
?
Y14=sqrt(47)*(284369571646197540+811049183285399890/k9+1593585237352147985/k18+1284787122351957913/k4+808867007855627582/k13+1055635222926786097/k22+876689391645197176/k8+599051054911901654/k17+717514008288053694/k3-160090977681806429/k12+1402330178847304956/k21+396563939508167961/k7+612829934225357570/k16+131453150804922790/k2+1510916177987207658/k11+514826183356996139/k20-274028959817533291/k6+873302494907324285/k15+432446621716420445/k1+550021082584940307/k10+1088241091554872128/k19+143212561542595904/k5);
?
Y15=sqrt(47)*(284369571646197540+811049183285399890/k8+1593585237352147985/k16+1284787122351957913/k1+808867007855627582/k9+1055635222926786097/k17+876689391645197176/k2+599051054911901654/k10+717514008288053694/k18-160090977681806429/k3+1402330178847304956/k11+396563939508167961/k19+612829934225357570/k4+131453150804922790/k12+1510916177987207658/k20+514826183356996139/k5-274028959817533291/k13+873302494907324285/k21+432446621716420445/k6+550021082584940307/k14+1088241091554872128/k22+143212561542595904/k7);
?
Y16=sqrt(47)*(284369571646197540+811049183285399890/k7+1593585237352147985/k14+1284787122351957913/k21+808867007855627582/k5+1055635222926786097/k12+876689391645197176/k19+599051054911901654/k3+717514008288053694/k10-160090977681806429/k17+1402330178847304956/k1+396563939508167961/k8+612829934225357570/k15+131453150804922790/k22+1510916177987207658/k6+514826183356996139/k13-274028959817533291/k20+873302494907324285/k4+432446621716420445/k11+550021082584940307/k18+1088241091554872128/k2+143212561542595904/k9);
?
Y17=sqrt(47)*(284369571646197540+811049183285399890/k6+1593585237352147985/k12+1284787122351957913/k18+808867007855627582/k1+1055635222926786097/k7+876689391645197176/k13+599051054911901654/k19+717514008288053694/k2-160090977681806429/k8+1402330178847304956/k14+396563939508167961/k20+612829934225357570/k3+131453150804922790/k9+1510916177987207658/k15+514826183356996139/k21-274028959817533291/k4+873302494907324285/k10+432446621716420445/k16+550021082584940307/k22+1088241091554872128/k5+143212561542595904/k11);
?
Y18=sqrt(47)*(284369571646197540+811049183285399890/k5+1593585237352147985/k10+1284787122351957913/k15+808867007855627582/k20+1055635222926786097/k2+876689391645197176/k7+599051054911901654/k12+717514008288053694/k17-160090977681806429/k22+1402330178847304956/k4+396563939508167961/k9+612829934225357570/k14+131453150804922790/k19+1510916177987207658/k1+514826183356996139/k6-274028959817533291/k11+873302494907324285/k16+432446621716420445/k21+550021082584940307/k3+1088241091554872128/k8+143212561542595904/k13);
?
Y19=sqrt(47)*(284369571646197540+811049183285399890/k4+1593585237352147985/k8+1284787122351957913/k12+808867007855627582/k16+1055635222926786097/k20+876689391645197176/k1+599051054911901654/k5+717514008288053694/k9-160090977681806429/k13+1402330178847304956/k17+396563939508167961/k21+612829934225357570/k2+131453150804922790/k6+1510916177987207658/k10+514826183356996139/k14-274028959817533291/k18+873302494907324285/k22+432446621716420445/k3+550021082584940307/k7+1088241091554872128/k11+143212561542595904/k15);
?
Y20=sqrt(47)*(284369571646197540+811049183285399890/k3+1593585237352147985/k6+1284787122351957913/k9+808867007855627582/k12+1055635222926786097/k15+876689391645197176/k18+599051054911901654/k21+717514008288053694/k1-160090977681806429/k4+1402330178847304956/k7+396563939508167961/k10+612829934225357570/k13+131453150804922790/k16+1510916177987207658/k19+514826183356996139/k22-274028959817533291/k2+873302494907324285/k5+432446621716420445/k8+550021082584940307/k11+1088241091554872128/k14+143212561542595904/k17);
?
Y21=sqrt(47)*(284369571646197540+811049183285399890/k2+1593585237352147985/k4+1284787122351957913/k6+808867007855627582/k8+1055635222926786097/k10+876689391645197176/k12+599051054911901654/k14+717514008288053694/k16-160090977681806429/k18+1402330178847304956/k20+396563939508167961/k22+612829934225357570/k1+131453150804922790/k3+1510916177987207658/k5+514826183356996139/k7-274028959817533291/k9+873302494907324285/k11+432446621716420445/k13+550021082584940307/k15+1088241091554872128/k17+143212561542595904/k19);
?
Y22=sqrt(47)*(284369571646197540+811049183285399890/k1+1593585237352147985/k2+1284787122351957913/k3+808867007855627582/k4+1055635222926786097/k5+876689391645197176/k6+599051054911901654/k7+717514008288053694/k8-160090977681806429/k9+1402330178847304956/k10+396563939508167961/k11+612829934225357570/k12+131453150804922790/k13+1510916177987207658/k14+514826183356996139/k15-274028959817533291/k16+873302494907324285/k17+432446621716420445/k18+550021082584940307/k19+1088241091554872128/k20+143212561542595904/k21);
?
(3)
F1/Y1^(1/23)=k21; F2/Y2^(1/23)=k4; F3/Y3^(1/23)=k15; F4/Y4^(1/23)=k10; F5/Y5^(1/23)=k12;
F6/Y6^(1/23)=k3; F7/Y7^(1/23)=k17; F8/Y8^(1/23)=k6; F9/Y9^(1/23)=k5; F10/Y10^(1/23)=k14; F11/Y11^(1/23)=k21;
?
最終得出：sin(2^u*pi/47) (u=1, 2, ……23)
· u=1
sin(2*pi/47)=(sqrt(47)+k21*Y1^(1/23)+k2*Y22^(1/23)+k4*Y2^(1/23)+k19*Y21^(1/23)+k15*Y3^(1/23)+k8*Y20^(1/23)+k10*Y4^(1/23)+k13*Y19^(1/23)+k12*Y5^(1/23)+k11*Y18^(1/23)+k3*Y6^(1/23)+k20*Y17^(1/23)+k17*Y7^(1/23)+k6*Y16^(1/23)+k6*Y8^(1/23)+k17*Y15^(1/23)+k5*Y9^(1/23)+k18*Y14^(1/23)+k14*Y10^(1/23)+k9*Y13^(1/23)+k21*Y11^(1/23)+k2*Y12^(1/23))/46;
?
· u=2
sin(4*pi/47)=(sqrt(47)+k20*Y1^(1/23)+k3*Y22^(1/23)+k2*Y2^(1/23)+k21*Y21^(1/23)+k12*Y3^(1/23)+k11*Y20^(1/23)+k6*Y4^(1/23)+k17*Y19^(1/23)+k7*Y5^(1/23)+k16*Y18^(1/23)+k20*Y6^(1/23)+k3*Y17^(1/23)+k10*Y7^(1/23)+k13*Y16^(1/23)+k21*Y8^(1/23)+k2*Y15^(1/23)+k19*Y9^(1/23)+k4*Y14^(1/23)+k4*Y10^(1/23)+k19*Y13^(1/23)+k10*Y11^(1/23)+k13*Y12^(1/23))/46;
?
……
也是一樣神奇的發(fā)現：u每加上1，2^u翻倍。這是X1^(1/23)前面的系數要除以k1，X2^(1/23)前面的系數要除以k2，X3^(1/23)前面的系數要除以k3，X4^(1/23)前面的系數要除以k4，……X22^(1/23)前面的系數要除以k22。后續(xù)的式子不列出，大家可根據這種規(guī)律推導出來。
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四．47次單位根舉例
利用前一章聲明的X值、本章聲明的Y值和23次單位根k1~k22，線性組合得到：
exp(2*pi*j/47)=(-1+X1^(1/23)+X22^(1/23)+X2^(1/23)+X21^(1/23)+k7*X3^(1/23)+k16*X20^(1/23)+k8*X4^(1/23)+k15*X19^(1/23)+k6*X5^(1/23)+k17*X18^(1/23)+k5*X6^(1/23)+k18*X17^(1/23)+k20*X7^(1/23)+k3*X16^(1/23)+k4*X8^(1/23)+k19*X15^(1/23)+k17*X9^(1/23)+k6*X14^(1/23)+k5*X10^(1/23)+k18*X13^(1/23)+k3*X11^(1/23)+k20*X12^(1/23)+j*(sqrt(47)+k21*Y1^(1/23)+k2*Y22^(1/23)+k4*Y2^(1/23)+k19*Y21^(1/23)+k15*Y3^(1/23)+k8*Y20^(1/23)+k10*Y4^(1/23)+k13*Y19^(1/23)+k12*Y5^(1/23)+k11*Y18^(1/23)+k3*Y6^(1/23)+k20*Y17^(1/23)+k17*Y7^(1/23)+k6*Y16^(1/23)+k6*Y8^(1/23)+k17*Y15^(1/23)+k5*Y9^(1/23)+k18*Y14^(1/23)+k14*Y10^(1/23)+k9*Y13^(1/23)+k21*Y11^(1/23)+k2*Y12^(1/23)))/46;
?
exp(92*pi*j/47)=(-1+X1^(1/23)+X22^(1/23)+X2^(1/23)+X21^(1/23)+k7*X3^(1/23)+k16*X20^(1/23)+k8*X4^(1/23)+k15*X19^(1/23)+k6*X5^(1/23)+k17*X18^(1/23)+k5*X6^(1/23)+k18*X17^(1/23)+k20*X7^(1/23)+k3*X16^(1/23)+k4*X8^(1/23)+k19*X15^(1/23)+k17*X9^(1/23)+k6*X14^(1/23)+k5*X10^(1/23)+k18*X13^(1/23)+k3*X11^(1/23)+k20*X12^(1/23)-j*(sqrt(47)+k21*Y1^(1/23)+k2*Y22^(1/23)+k4*Y2^(1/23)+k19*Y21^(1/23)+k15*Y3^(1/23)+k8*Y20^(1/23)+k10*Y4^(1/23)+k13*Y19^(1/23)+k12*Y5^(1/23)+k11*Y18^(1/23)+k3*Y6^(1/23)+k20*Y17^(1/23)+k17*Y7^(1/23)+k6*Y16^(1/23)+k6*Y8^(1/23)+k17*Y15^(1/23)+k5*Y9^(1/23)+k18*Y14^(1/23)+k14*Y10^(1/23)+k9*Y13^(1/23)+k21*Y11^(1/23)+k2*Y12^(1/23)))/46;
?
exp(4*pi*j/47)=(-1+k22*X1^(1/23)+k1*X22^(1/23)+k21*X2^(1/23)+k2*X21^(1/23)+k4*X3^(1/23)+k19*X20^(1/23)+k4*X4^(1/23)+k19*X19^(1/23)+k1*X5^(1/23)+k22*X18^(1/23)+k22*X6^(1/23)+k1*X17^(1/23)+k13*X7^(1/23)+k10*X16^(1/23)+k19*X8^(1/23)+k4*X15^(1/23)+k8*X9^(1/23)+k15*X14^(1/23)+k18*X10^(1/23)+k5*X13^(1/23)+k15*X11^(1/23)+k8*X12^(1/23)+j*(sqrt(47)+k20*Y1^(1/23)+k3*Y22^(1/23)+k2*Y2^(1/23)+k21*Y21^(1/23)+k12*Y3^(1/23)+k11*Y20^(1/23)+k6*Y4^(1/23)+k17*Y19^(1/23)+k7*Y5^(1/23)+k16*Y18^(1/23)+k20*Y6^(1/23)+k3*Y17^(1/23)+k10*Y7^(1/23)+k13*Y16^(1/23)+k21*Y8^(1/23)+k2*Y15^(1/23)+k19*Y9^(1/23)+k4*Y14^(1/23)+k4*Y10^(1/23)+k19*Y13^(1/23)+k10*Y11^(1/23)+k13*Y12^(1/23)))/46;
?
exp(90*pi*j/47)=(-1+k22*X1^(1/23)+k1*X22^(1/23)+k21*X2^(1/23)+k2*X21^(1/23)+k4*X3^(1/23)+k19*X20^(1/23)+k4*X4^(1/23)+k19*X19^(1/23)+k1*X5^(1/23)+k22*X18^(1/23)+k22*X6^(1/23)+k1*X17^(1/23)+k13*X7^(1/23)+k10*X16^(1/23)+k19*X8^(1/23)+k4*X15^(1/23)+k8*X9^(1/23)+k15*X14^(1/23)+k18*X10^(1/23)+k5*X13^(1/23)+k15*X11^(1/23)+k8*X12^(1/23)-j*(sqrt(47)+k20*Y1^(1/23)+k3*Y22^(1/23)+k2*Y2^(1/23)+k21*Y21^(1/23)+k12*Y3^(1/23)+k11*Y20^(1/23)+k6*Y4^(1/23)+k17*Y19^(1/23)+k7*Y5^(1/23)+k16*Y18^(1/23)+k20*Y6^(1/23)+k3*Y17^(1/23)+k10*Y7^(1/23)+k13*Y16^(1/23)+k21*Y8^(1/23)+k2*Y15^(1/23)+k19*Y9^(1/23)+k4*Y14^(1/23)+k4*Y10^(1/23)+k19*Y13^(1/23)+k10*Y11^(1/23)+k13*Y12^(1/23)))/46;
?
……
剩下的42個復數解也能根據k值得變化規(guī)律推理得到。
